surreptitious75 wrote:Nothing north of the north pole is only true in relation to the Earth as an isolated body but not so in relation to its position within the UniverseSo this is where it fails as an analogy in trying to explain why time did not exist before the Big Bang as it assumes nothing existed before it did

If there is no time, then what does "before" mean?

It is not known what did or did not exist before the Big Bang because that is only as far as back as physics can currently goThis is demonstrably not the same as saying it cannot go any further back because the BB is the absolute beginning of time

If time had a beginning, then there is no before. If there were a before, then after the before would not be the beginning of time.

If the singularity was a space of zero volume and infinite density then nothing could have existed before it including time

This definition is wrong because both zero volume and infinite density cannot exist in actualityAs the former would have no dimension or property and the latter can only exist in finite form

A singularity less absolute in physicality however would allow for time to exist before it as it would not be the totality of all that existed

Oh I see. But in order to have time, we need things in motion through a spacial construct. How can there be things in motion if the universe is so small?

Also if time did begin at the Big Bang it would mean absolute nothing existed before it but this is actually invalidated by quantum mechanics

But time doesn't exist relative to light, yet there is not nothing.

As absolute nothing can only exist infinitesimally not infinitely because of the existence of quantum fluctuations which disturb vacuum states

So quantum mechanics absolutely forbids the existence of a singularity as traditionally defined

Yes that makes sense.

Also the Big Bang was not the beginning of the Universe as such but only local cosmic expansion

Thanks for the link and I would have showcased it all on its own had I seen it first

The point I am making is something different. I am pointing out that:

All of our modern theories of physics rely ultimately on highly abstract infinitary mathematics

That doesn't mean that they necessarily do; only that so far, that's how the history has worked out.

I see what you mean, but as Max pointed out when describing air as seeming continuous while actually being discrete, it's easier to model a continuum than a bazillion molecules, each with functional probabilistic movements of their own. Essentially, it's taking an average and it turns out that it's pretty accurate.

But what I was saying previously is that we work with the presumed ramifications of infinity, "as if" this or that were infinite, without actually ever using infinity itself. For instance, y = 1/x as x approaches infinity, then y approaches 0, but we don't actually USE infinity in any calculations, but we extrapolate.

There is at the moment no credible alternative. There are attempts to build physics on constructive foundations (there are infinite objects but they can be constructed by algorithms). But not finitary principles, because to do physics you need the real numbers; and to construct the real numbers we need infinite sets.

Hilbert pointed out there is a difference between boundless and infinite. For instance space is boundless as far as we can tell, but it isn't infinite in size and never will be until eternity arrives. Why can't we use the boundless assumption instead of full-blown infinity?

1) The rigorization of Newton's calculus culminated with infinitary set theory.

Newton discovered his theory of gravity using calculus, which he invented for that purpose.

I didn't know he developed calculus specifically to investigate gravity. Cool! It does make sense now that you mention it.

However, it's well-known that Newton's formulation of calculus made no logical sense at all. If \(\Delta y\) and \(\Delta x\) are nonzero, then \(\frac{\Delta y}{\Delta x}\) isn't the derivative. And if they're both zero, then the expression makes no mathematical sense! But if we pretend that it does, then we can write down a simple law that explains apples falling to earth and the planets endlessly falling around the sun.

I'm going to need some help with this one. If dx = 0, then it contains no information about the change in x, so how can anything result from it? I've always taken dx to mean a differential that is smaller than can be discerned, but still able to convey information. It seems to me that calculus couldn't work if it were based on division by zero, and that if it works, it must not be. What is it I am failing to see? I mean, it's not an issue of 0/0 making no mathematical sense, it's a philosophical issue of the nonexistence of significance because there is nothing in zero to be significant.

2) Einstein's gneral relativity uses Riemann's differential geometry.

In the 1840's Bernhard Riemann developed a general theory of surfaces that could be Euclidean or very far from Euclidean. As long as they were "locally" Euclidean. Like spheres, and torii, and far weirder non-visualizable shapes. Riemann showed how to do calculus on those surfaces. 60 years later, Einstein had these crazy ideas about the nature of the universe, and the mathematician Minkowski saw that Einstein's ideas made the most mathematical sense in Riemann's framework. This is all abstract infinitary mathematics.

Isn't this the same problem as previous? dx=0?

3) Fourier series link the physics of heat to the physics of the Internet; via infinite trigonometric series.

In 1807 Joseph Fourier analyzed the mathematics of the distribution of heat through an iron bar. He discovered that any continuous function can be expressed as an infinite trigonometric series, which looks like this:

I only posted that because if you managed to survive high school trigonometry, it's not that hard to unpack. You're composing any motion into a sum of periodic sine and cosine waves, one wave for each whole number frequency. And this is an infinite series of real numbers, which we cannot make sense of without using infinitary math.

I can't make sense of it WITH infinitary math lol! What's the cosine of infinity? What's the infnite-th 'a'?

4) Quantum theory is functional analysis.

If you took linear algebra, then functional analysis can be thought of as infinite-dimensional linear algebra combined with calculus. Functional analysis studies spaces whose points are actually functions; so you can apply geometric ideas like length and angle to wild collections of functions. In that sense functional analysis actually generalizes Fourier series.

Quantum mechanics is expressed in the mathematical framework of functional analysis. QM takes place in an infinite-dimensional Hilbert space. To explain Hilbert space requires a deep dive into modern infinitary math. In particular, Hilbert space is complete, meaning that it has no holes in it. It's like the real numbers and not like the rational numbers.

QM rests on the mathematics of uncountable sets, in an essential way.

Well, thanks to Hilbert, I've already conceded that the boundless is not the same as the infinite and if it were true that QM required infinity, then no machine nor human mind could model it. It simply must be true that open-ended finites are actually employed and underpin QM rather than true infinite spaces.

Like Max said, "Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations."

We can *claim* physics is based on infinity, but I think it's more accurate to say *pretend* or *fool ourselves* into thinking such.

Max continued with, "Our challenge as physicists is to discover this elegant way and the infinity-free equations describing it—the true laws of physics. To start this search in earnest, we need to question infinity. I’m betting that we also need to let go of it."

He said, "let go of it" like we're clinging to it for some reason external to what is true. I think the reason is to be rid of god, but that's my personal opinion. Because if we can't have infinite time, then there must be a creator and yada yada. So if we cling to infinity, then we don't need the creator. Hence why Craig quotes Hilbert because his first order of business is to dispel infinity and substitute god.

I applaud your effort, I really do, and I've learned a lot of history because of it, but I still cannot concede that infinity underpins anything and I'd be lying if I said I could see it. I'm not being stubborn and feel like I'm walking on eggshells being as amicable and conciliatory as possible in trying not to offend and I'm certainly ready to say "Ooooohhh... I see now", but I just don't see it.

ps -- There's our buddy Hilbert again. He did many great things. William Lane Craig misuses and abuses Hilbert's popularized example of the infinite hotel to make disingenuous points about theology and in particular to argue for the existence of God. That's what I've got against Craig.

Craig is no friend of mine and I was simply listening to a debate on youtube (I often let youtube autoplay like a radio) when I heard him quote Hilbert, so I dug into it and posted what I found. I'm not endorsing Craig lol

5) Cantor was led to set theory from Fourier series.

In every online overview of Georg Cantor's magnificent creation of set theory, nobody ever mentions how he came upon his ideas. It's as if he woke up one day and decided to revolutionize the foundations of math and piss off his teacher and mentor Kronecker. Nothing could be further from the truth.

Cantor was in fact studing Fourier's trigonometric series! One of the questions of that era was whether a given function could have more than one distinct Fourier series. To investigate this problem, Cantor had to consider the various types of sets of points on which two series could agree; or equivalently, the various sets of points on which a trigonometric series could be zero. He was thereby led to the problem of classifying various infinite sets of real numbers; and that led him to the discovery of transfinite ordinal and cardinal numbers. (Ordinals are about order in the same way that cardinals are about quantity).

I still can't understand how one infinity can be bigger than another since, to be so, the smaller infinity would need to have limits which would then make it not infinity.

In other words, and this is a fact that you probably will not find stated as clearly as I'm stating it here:

If you begin by studying the flow of heat through an iron rod; you will inexorably discover transfinite set theory.

Right, because of what Max said about the continuum model vs the actual discrete. Heat flow is actually IR light flow which is radiation from one molecule to another: a charged particle vibrates and vibrations include accelerations which cause EM radiation that emanates out in all directions; then the EM wave encounters another charged particle which causes vibration and the cycle continues until all the energy is radiated out. It's a discrete process from molecule to molecule, but is modeled as continuous for simplicity's sake.

I've long taken issue with the 3 modes of heat transmission (conduction, convention, radiation) because there is only radiation. Atoms do not touch, so they can't conduct, but the van der waals force simply transfers the vibrations more quickly when atoms are sufficiently close. Convection is simply vibrating atoms in linear motion that are radiating IR light. I have many issues with physics and have often described it as more of an art than a science (hence why it's so difficult). I mean, there are pages and pages on the internet devoted to simply trying to define heat.

Physics is a mess. What gamma rays are, depends who you ask. They could be high-frequency light or any radiation of any frequency that originated from a nucleus. But I'm digressing....

I do not know what that means in the ultimate scheme of things. But I submit that even the most ardent finitist must at least give consideration to this historical reality.

It just means we're using averages rather than discrete actualities and it's close enough.

I hope I've been able to explain why I completely agree with your point that infinities in physical equations don't imply the actual existence of infinities. Yet at the same time, I am pointing out that our best THEORIES of physics are invariably founded on highly infinitary math. As to what that means ... for my own part, I can't help but feel that mathematical infinity is telling us something about the world. We just don't know yet what that is.

I think it means there are really no separate things and when an aspect of the universe attempts to inspect itself in order to find its fundamentals or universal truths, it will find infinity like a camera looking at its own monitor. Infinity is evidence of the continuity of the singular universe rather than an existing truly boundless thing. Infinity simply means you're looking at yourself.

Anyway, great post! Please don't be mad. Everyone here values your presence and are intimidated by your obvious mathematical prowess Don't take my pushback too seriously I'd prefer if we could collaborate as colleagues rather than competing.

Serendipper wrote:How can there be things in motion if the universe is so small ?

Small is a relative term and from a classical perspective the quantum is incredibly so but this does not mean motion is not actually possible at this level Three of the four fundamental forces operate at the quantum level inside of an atom in relation to the motion of its protons and neutrons and electrons Also motion is a universal constant so any space no matter how small will feature motion even if it is just quantum fluctuations disturbing vacuum states

Serendipper wrote:I still cant understand how one infinity can be bigger than another since to be so the smaller infinity would need to have limits which would then make it not infinity

This statement violates the Law Of Non Contradiction as it is impossible for an infinity to also be not infinity The problem here is that in relation to small infinity you are focusing on the small rather than the infinity

All infinite sets contain an infinite number of members but some of them will have more members than others and some will have lessFor example the infinite set of primes is smaller than the infinite set of reals because primes occur less frequently on the number line

But primes are no less infinite than the reals despite this [ and yes it is incredibly counter intuitive but more importantly it is also true ]

Serendipper wrote:How can there be things in motion if the universe is so small ?

Small is a relative term and from a classical perspective the quantum is incredibly so but this does not mean motion is not actually possible at this level Three of the four fundamental forces operate at the quantum level inside of an atom in relation to the motion of its protons and neutrons and electrons Also motion is a universal constant so any space no matter how small will feature motion even if it is just quantum fluctuations disturbing vacuum states

I think we first need to answer the question of if there is a smallest point in the universe.

If there is a smallest point, then the extents of the universe would have to be bigger than that smallest point in order to have concepts of moving from point to point.

If there is not a smallest point, then the universe has no size relative to its smallest point, and regardless how small the universe became, it would still be infinitely larger than its own points.

The evidence so far suggests that the smaller we go, the less certain we can be of location and the more probabilistic location becomes. I am under the impression that space is not empty space with things moving about independent of the space-substance, but a collection of fields analogous to a computer screen where things cannot be smaller than the pixel (a quantum) that generates them. So if the universe were smaller than a pixel, then how could movement exist?

Also, neither time nor space exists relative to a photon. Every photon emission and reception from the beginning of time to the end of time, took no time and covered no space from its own perspective, which seems to imply there is no objective time standard by which to measure spacetime itself, but that time is an artifact of spacetime and contained only therein. Time is subject to and emergent from spacetime and not objective to it.

It seems obvious that something existed before, but I can't understand what "before" means. From the photon's perspective, where was it yesterday? "Yesterday" only applies to me and not the photon which itself caused a sense of "yesterday" to manifest. Time cannot apply to the thing that causes time. It's like trying to use a knife to cut the same knife.

Serendipper wrote:I still cant understand how one infinity can be bigger than another since to be so the smaller infinity would need to have limits which would then make it not infinity

This statement violates the Law Of Non Contradiction as it is impossible for an infinity to also be not infinity The problem here is that in relation to small infinity you are focusing on the small rather than the infinity

All infinite sets contain an infinite number of members but some of them will have more members than others and some will have lessFor example the infinite set of primes is smaller than the infinite set of reals because primes occur less frequently on the number line

But primes are no less infinite than the reals despite this [ and yes it is incredibly counter intuitive but more importantly it is also true ]

Primes are infinite only because reals are infinite. If reals were not, then primes could not be, so the cardinality of primes is dependent upon the cardinality of reals. This is like saying the age of a son will always be less than the age of his father, except if they live forever, in which case they will be the same age on the day that forever arrives (which can never happen). It's the same with primes and reals: there will always be more reals than primes until the day eternity arrives.

If a completed infinity of reals are ALL numbers in actual existence and potential existence and nonexistence and any other existence you can conceive, then from where will you find more numbers to have a larger infinity? And if you do find more numbers to make a larger infinity, then it wasn't ALL the numbers to start with.

Ordinals assume an infinity of reals can be completed and then some new number comes in order right after that completed infinity. Like Gauss, I can't make sense of a completed infinity. Further, the first infinity should include all of the second or else it wasn't infinity.

Serendipper wrote:I think we first need to answer the question of if there is a smallest point in the universe

This is the point that cannot be made any smaller without it becoming non existent / non physicalHowever this is not a real point that can actually be measured either physically or mathematically In mathematical terms this would be 0 . 00000 .... extending to infinity with I added on to the end

A number as close to 0 as it is possible to be without it actually being 0 and because this number is infinite then it cannot ever be expressed in finite termsThis is the same problem with the physical for no matter how small you go you can always go smaller without ever reaching the non existent / non physical

surreptitious75 wrote:In mathematical terms this would be 0 . 00000 .... extending to infinity with I added on to the end

There is no such thing. First, if x is a positive real number, then x/2 is a smaller one so x wasn't the smallest; and since x was an arbitrary positive real, there is no smallest positive real.

Secondly, the meaning of a decimal string is that each digit position represents a negative integer power of 10. So .1234 means 1/10 + 2/100 + 3/1000 + 4/10000, and likewise for infinite decimals. There is no corresponding interpretation of .000...1 where there are infinitely many 0's preceding the 1. It's a meaningless expression.

Serendipper wrote:Anyway, great post! Please don't be mad.

Thank you for the kind words. Mad as in angry? Or mad as in crazy? Haha.

I don't mean to intimidate anyone or show off in any way. I just try to put things into their proper mathematical context. For example a lot of people have an intuition that there's a smallest positive real number, but I just gave two demonstrations that this cannot possibly be the case.

Moreover, there is a philosophical reason as well. The real numbers are a continuum. If we visualize them as a string of bowling balls, they're not a continuum.

Serendipper wrote: I'd prefer if we could collaborate as colleagues rather than competing.

Yes definitely.

Serendipper wrote:We can *claim* physics is based on infinity, but I think it's more accurate to say *pretend* or *fool ourselves* into thinking such.

Just for sake of discussion: this is the kind of thing that makes me mad angry AND mad crazy. I never said that physics is based on mathematical infinity. I said, and I will ask you to go back and verify this for yourself, that all of our admittedly contingent THEORIES of physics are based on mathematical infinity. That is a fact of the world that anyone can verify, and in support of which I gave copious references.

And if you don't know the difference between physics and THEORIES of physics, it's this. If you are sitting down right now and gravity disappears, you will float up to the ceiling of the room you're in. But if our THEORIES of gravity all disappear, you will still be stuck to the earth. The only difference is that you won't be able to go to school to learn what people think about WHY that is. Do you see the difference?

We are not fooling ourselves that our theories are based on mathematical infinity. That's a matter of fact, verifiable from anything you can read online or in books.

Are you willing to acknowledge that you profoundly mischaracterized what I said? I'd like to have a congenial conversation too, but that's not possible when you deliberately claim I said things I did not say. I went to great lengths to point out that I was talking about our THEORIES of physics, and not at all how the world itself may happen to work. Did I not make that clear?

ps -- I'm not upset. You don't have to walk on eggshells. My point is very clear in my own mind but evidently I'm not making my case! I'll work harder.

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Serendipper wrote:Primes are infinite only because reals are infinite.

Interestingly, not so much. Euclid demonstrated the infinitude of primes two thousand years before we had any concept of the real numbers. And in fact Euclid's original proof is valid even without any theory of the real numbers today. We do not need to assume the axiom of infinity to prove that there is no largest prime. In other words if there are no infinite sets at all, it's still true that there's no largest prime, and Euclid's proof goes through in its original form.

Serendipper wrote:But what I was saying previously is that we work with the presumed ramifications of infinity, "as if" this or that were infinite, without actually ever using infinity itself. For instance, y = 1/x as x approaches infinity, then y approaches 0, but we don't actually USE infinity in any calculations, but we extrapolate.

QM is based on an infinite-dimensional vector space called Hilbert space. It's infinite-dimensional. Literally and truly, Whether the world itself is, or only our theory is, I can't say. But QM takes place in infinite-dimensional space and that's a fact. I gave a link to Hilbert space earlier.

Serendipper wrote:I'm going to need some help with this one. If dx = 0, then it contains no information about the change in x, so how can anything result from it? I've always taken dx to mean a differential that is smaller than can be discerned, but still able to convey information.

Perhaps that's more of Leibniz's view or more of a philosophical view. In calculus, the derivative is the limit of deltaY/deltaX as deltaX goes to zero. What that "means" is a bit elusive but one gets used to it after a while.

Serendipper wrote: It seems to me that calculus couldn't work if it were based on division by zero, and that if it works, it must not be.

Yes, Berkeley made the same point, calling derivatives the "ghosts of departed quantities." Today we have the theory of limits that makes it all mathematically rigorous; though I agree that it's perhaps not entirely satisfying philosophically.

Serendipper wrote:What is it I am failing to see?

The modern theory of limits.

Serendipper wrote: I mean, it's not an issue of 0/0 making no mathematical sense, it's a philosophical issue of the nonexistence of significance because there is nothing in zero to be significant.

People have been arguing about this since Newton and it's still going on. There are modern incarnations of infinitesimals.

Serendipper wrote:Well, thanks to Hilbert, I've already conceded that the boundless is not the same as the infinite and if it were true that QM required infinity, then no machine nor human mind could model it.

Hilbert space is infinite-dimensional and that's what they use to do QM.

Serendipper wrote:I applaud your effort, I really do, and I've learned a lot of history because of it

I appreciate your kinds words. it is a FACT that all of our theories of physics require actual infinity. Not unboundedness. Actual infinities. Infinities of points in a continuous space in which to do relativity; infinities of wave functions in an infinite-dimensional Hilbert space to do QM. Nobody has a finitary theory of physics. Nobody.

Serendipper wrote:but I still cannot concede that infinity underpins anything and I'd be lying if I said I could see it.

I wonder if you are still conflating physics (how the world is) with theories of physics (our mathematical models). Could that be the case?

Serendipper wrote: I'm not being stubborn and feel like I'm walking on eggshells being as amicable and conciliatory as possible in trying not to offend and I'm certainly ready to say "Ooooohhh... I see now", but I just don't see it.

Perfectly ok. I'm easily provoked but I get over it quickly. Your philosophical points are valid but the theories themselves require mathematical infinities. Are you saying you don't believe our theories themselves require mathematical infinity? But they do. I've given as many explanations as I can think of.

About the 0/0. Say you're driving in your car and your speedometer says 50mph. That's the derivative of your position function. Conceptually it says that if we divide your distance over time for smaller and smaller intervals, the "instantaneous" change in position is 50. But if we could stop time in that one instant, where is the motion? Back to Zeno again. Ancient mystery. And yet your speedometer has no problem measuring your speed.

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WTF wrote:If you are sitting down right now and gravity disappears you will float up to the ceiling of the room you are in. But if our THEORIES of gravity all disappear you will still be stuck to the earth. The only difference is that you will not be able to go to school to learn what people think about WHY that is

The map must never be confused for the territory because it is merely an approximation of it and they are therefore not the same

WTF wrote:If you are sitting down right now and gravity disappears you will float up to the ceiling of the room you are in. But if our THEORIES of gravity all disappear you will still be stuck to the earth. The only difference is that you will not be able to go to school to learn what people think about WHY that is

The map must never be confused for the territory because it is merely an approximation of it and they are therefore not the same

But of course. And I went to great lengths to explain that I am talking about the maps and NOT the territory. Every THEORY of physics is based on mathematical infinity.

You know I wonder if I'm making a point of far greater subtlety that I realize, because now two people are pushing back for (to my mind) no reason.

I agree that I have no idea how the world "really" is, or if the question is even meaningful. But our mathematical theories of physics, the historically contingent theories of Galileo and Kepler and Newton and Lagrange and Maxwell and Einstein and Witten and all these people ... these THEORIES are all expressed using infinitary mathematics.

I know nothing of the territory. But the MAPS are based on infinitary mathematics.

WTF wrote:For example a lot of people have an intuition that there is a smallest positive real number but I just gave two demonstrations that this cannot possibly be the caseMoreover there is a philosophical reason as well . The real numbers are a continuum . If we visualize them as a string of bowling balls they are not a continuum

I like this analogy even though the continuum or infinite set is made up of infinite individual members like all infinite sets

Serendipper wrote:I think we first need to answer the question of if there is a smallest point in the universe

This is the point that cannot be made any smaller without it becoming non existent / non physical

Well, it's like the north pole. The north pole is the place where we cannot go any farther north. So the smallest point is the point where it doesn't make sense to go smaller because there is nowhere to go. Like my pixel analogy.

However this is not a real point that can actually be measured either physically or mathematically In mathematical terms this would be 0 . 00000 .... extending to infinity with I added on to the endA number as close to 0 as it is possible to be without it actually being 0 and because this number is infinite then it cannot ever be expressed in finite terms

It's not a problem of precision, but that there is no there, there.

Physical limits are not like a government insisting we stop, but we cannot go faster than light because the universe would have negative size, which doesn't make sense. I mean, you could arrive before you left and see yourself off. Size is similar. There can't be anything smaller than the thing that determines size. Just like there is no standard of temporal measurement applicable to light because light defines time.

This is the same problem with the physical for no matter how small you go you can always go smaller without ever reaching the non existent / non physical

Math transcends reality.

When you hear this, you may stop and think, “Surely, if I have a length, then I half it, and I repeat this over and over, I will be able to get to something smaller.” However, this is an occasion where physics doesn’t allow something that mathematics does. For example, think about moving faster than the speed of light. On paper you could apply a force to a mass and accelerate it up and past the speed of light, but we know that in nature that just is not physically possible because the mass of the object (and thus, the energy needed to speed it up) goes towards infinity—both keep growing without any limit. So what we can do on paper, we can’t do in reality.

So, how does a tiny number such as this tie into physics? If two particles were separated by the Planck length, or anything less, then it is impossible to actually tell their positions apart. Moreover, any effects of quantum gravity at this scale (if there are any) are entirely unknown as space itself is not properly defined. In a sense, you could say that, even if we were to develop methods of measurements that took us down to these scales, we would never be able to measure anything smaller despite any sort of improvements to our equipment or methods.https://futurism.com/apotd-ngc-1316-2

It dawned on me while talking to James over a year ago that it didn't makes sense to have an infinity of smaller particles, but putting the reason why into words is hard. Essentially there would be no reference point, no anchor to any kind of reality. If the size of an atom could have been any of the infinity of sizes, then why this one? It's just too arbitrary on the hierarchy, but if there is a smallest size, then it makes total sense why an atom is the size it is because it's relative to that smallest size.

Also, if there is an infinite hierarchy of possible sizes, then we need an external standard of measurement to measure what size an atom is because there would be no way to tell how big it would be relative to the universe itself. But if there is a smallest size in the universe, then no such external measurement is required to define the size of an atom (although the size of the smallest point becomes arbitrary relative to an external standard of measurement). Is this making sense? IOW, we need a zero point for an origin in order to have anything.

Infinite computer memory is another analogy. Once something is stored in such memory, how could it be found again?

Serendipper wrote:Infinite computer memory is another analogy. Once something is stored in such memory, how could it be found again?

I see no problem at all. As in an actual computer, each mem location is labeled with a positive integer address: 1, 2, 3, 4, ..., and we store and fetch data to and from a given location via its address. How would this change if we simply used all the integers? We store value x at location n; and later, we fetch value x from location n.

In fact a Turing machine is conceptually just like this, and it does not require infinite memory; only unbounded memory. We don't require that there are an actual infinity of memory cells (or locations on a paper tape, as Turing puts it). Given some positive integer n, the tape has cell n. There are never an actual infinity of cells; only an unbounded array.

Serendipper wrote:Infinite computer memory is another analogy. Once something is stored in such memory, how could it be found again?

I see no problem at all. As in an actual computer, each mem location is labeled with a positive integer address: 1, 2, 3, 4, ..., and we store and fetch data to and from a given location via its address. How would this change if we simply used all the integers? We store value x at location n; and later, we fetch value x from location n.

Because infinite memory would be a box with no walls and therefore have no objective reference point for where to begin addressing.

I think we have to be careful not to conflate the unbounded finite with the completed infinite.

If we had infinite cells, which one would we label the zero? Zero has no objective anchor and must be chosen at random. Whichever cell we randomly choose for zero will have infinite cells under and infinite cells above it, so any zero isn't a true zero, but an arbitrary location to start sequentially ordering. It's not finite on one end and infinite on the other, but it can only be infinite in both directions because if the idea is to assume that we can always add one, then the zero must itself be a product of that addition, as is every integer.

Half of infinite capacity is absurd to me like a squared circle. Integers extend in both directions, but we cut it in half at zero and pretend we've corralled the infinite as if we could somehow hold the end of an infinite rope. We can conceptually cut it in half all we want, but we can't remove the fact that the negative integers still exist and that zero was merely an arbitrary starting point where we decided to cut it. We could cut it at 5 and say that's the beginning. Or 437859348 and say that's the beginning. Or -53478. Or any number. Zero is an arbitrary delineation on an infinite continuum that has neither beginning nor end and is not an objective anchor to actuality, and in actuality we'd be presented with a completed infinite thing and tasked with ordering it rather than sequentially building an infinite thing by adding one forever. Like, if space were infinite, where is the finite edge where all that infinite space begins?

A finite stick of memory can be ordered because it has an edge, but infinite sticks have no edges and starting points must be chosen at random, then when the computer returns for the information, it must choose another random point and it has 1 out of infinity chance of finding the same point as before, so there is no chance it could find the info it deposited. Infinite memory is zero memory: the absence by ubiquitousness.

Let's say there is infinite time and here we are in the middle of infinite past and infinite future, what is this location in time referenced to? If time is finite with a beginning, then we would just reference this point to the objective beginning, but in infinite time, we could only reference to another arbitrary point that itself is deficient of objective reference. No point in infinite time has distinction from any other. It's the same for infinite particle sizes.

If particle sizes are arbitrary and there are an infinity of smaller ones, then obviously there are infinite smaller universes just like this one, where you and I are having the same conversation. And if time were infinite, then we've already had this conversation an infinite number of times and are bound to keep repeating it forever. Anything that has any probability of happening, must happen infinite times in infinite time. Since we know for sure that this universe happened, then it will happen infinite times on infinite particle-size levels. That's a little bit ridiculous for me lol. It's easier for me to believe the answer is deeper and we haven't found it yet than to be satisfied with infinity as a solution with all its absurd ramifications.

In fact a Turing machine is conceptually just like this, and it does not require infinite memory; only unbounded memory. We don't require that there are an actual infinity of memory cells (or locations on a paper tape, as Turing puts it). Given some positive integer n, the tape has cell n. There are never an actual infinity of cells; only an unbounded array.

You have literally described a TM.

Unbounded memory will always be finite memory that hasn't yet found a bound (like the money supply is theoretically unbounded but always finite). Infinite memory is the impossibility of a bound and yet in some sort of completed form (a wall-less box).

surreptitious75 wrote:In mathematical terms this would be 0 . 00000 .... extending to infinity with I added on to the end

There is no such thing. First, if x is a positive real number, then x/2 is a smaller one so x wasn't the smallest; and since x was an arbitrary positive real, there is no smallest positive real.

Secondly, the meaning of a decimal string is that each digit position represents a negative integer power of 10. So .1234 means 1/10 + 2/100 + 3/1000 + 4/10000, and likewise for infinite decimals. There is no corresponding interpretation of .000...1 where there are infinitely many 0's preceding the 1. It's a meaningless expression.

We can't have 10^-x where x -> infinity?

I know what he means though.

Serendipper wrote:Anyway, great post! Please don't be mad.

Thank you for the kind words. Mad as in angry? Or mad as in crazy? Haha.

Yeah I know, but some here consider themselves fairly proficient in math when measured against their peers and then you come along and wtf lol

I just try to put things into their proper mathematical context. For example a lot of people have an intuition that there's a smallest positive real number, but I just gave two demonstrations that this cannot possibly be the case.

True, but is there a smallest number that you could write down? I think that's the point of all this: where the abstract meets the reality. Like the smallest point, the largest size, the fastest speed,,, math transcends all of it.

Moreover, there is a philosophical reason as well. The real numbers are a continuum. If we visualize them as a string of bowling balls, they're not a continuum.

True, but in reality there are smallest quantifiables which makes it discrete, even though the quantums are continuous with other quantums in actuality, but it's not possible to differentiate a smaller one.

Serendipper wrote:We can *claim* physics is based on infinity, but I think it's more accurate to say *pretend* or *fool ourselves* into thinking such.

Just for sake of discussion: this is the kind of thing that makes me mad angry AND mad crazy. I never said that physics is based on mathematical infinity. I said, and I will ask you to go back and verify this for yourself, that all of our admittedly contingent THEORIES of physics are based on mathematical infinity. That is a fact of the world that anyone can verify, and in support of which I gave copious references.

Well: 1) it was really just a shorthand "manner of speaking" like x approaches infinity. 2) To me, physics is the theories. Physics is not the reality just like words are not things. That's just how I set it up in my mind. 3) You did it yourself: "I know of nobody who claims to be able to found physics on finitary principles." viewtopic.php?f=4&t=194376&start=175#p2713785

And if you don't know the difference between physics and THEORIES of physics, it's this. If you are sitting down right now and gravity disappears, you will float up to the ceiling of the room you're in. But if our THEORIES of gravity all disappear, you will still be stuck to the earth. The only difference is that you won't be able to go to school to learn what people think about WHY that is. Do you see the difference?

Yes I see, you are equating "physics" with the actuality while I'm equating it with the knowledge of actuality.

We are not fooling ourselves that our theories are based on mathematical infinity.

When I say "fooling ourselves" what I mean is a mind or machine cannot conceptualize or imagine the infinite. It can't be modeled, but we imagine the biggest thing we can, then extrapolate by inference and call it good enough. We assume there are bigger numbers without actually empirically verifying it, and the assumption that something exists without proof is what I mean by "fooling ourselves".

That's a matter of fact, verifiable from anything you can read online or in books.

But Max said:

Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite. So if we can do without infinity to figure out what happens next, surely nature can, too—in a way that’s more deep and elegant than the hacks we use for our computer simulations.

Are you willing to acknowledge that you profoundly mischaracterized what I said? I'd like to have a congenial conversation too, but that's not possible when you deliberately claim I said things I did not say. I went to great lengths to point out that I was talking about our THEORIES of physics, and not at all how the world itself may happen to work. Did I not make that clear?

ps -- I'm not upset. You don't have to walk on eggshells. My point is very clear in my own mind but evidently I'm not making my case! I'll work harder.

Serendipper wrote:Primes are infinite only because reals are infinite.

Interestingly, not so much. Euclid demonstrated the infinitude of primes two thousand years before we had any concept of the real numbers. And in fact Euclid's original proof is valid even without any theory of the real numbers today. We do not need to assume the axiom of infinity to prove that there is no largest prime. In other words if there are no infinite sets at all, it's still true that there's no largest prime, and Euclid's proof goes through in its original form.

If reals are not infinite, then primes cannot be infinite, right? So primes are infinite only because reals are infinite and the infinity of primes is just a consequence of the infinity of reals.

Serendipper wrote:But what I was saying previously is that we work with the presumed ramifications of infinity, "as if" this or that were infinite, without actually ever using infinity itself. For instance, y = 1/x as x approaches infinity, then y approaches 0, but we don't actually USE infinity in any calculations, but we extrapolate.

QM is based on an infinite-dimensional vector space called Hilbert space. It's infinite-dimensional. Literally and truly, Whether the world itself is, or only our theory is, I can't say. But QM takes place in infinite-dimensional space and that's a fact. I gave a link to Hilbert space earlier.

Well where is infinity actually used in a calculation?

Serendipper wrote:I'm going to need some help with this one. If dx = 0, then it contains no information about the change in x, so how can anything result from it? I've always taken dx to mean a differential that is smaller than can be discerned, but still able to convey information.

Perhaps that's more of Leibniz's view or more of a philosophical view. In calculus, the derivative is the limit of deltaY/deltaX as deltaX goes to zero. What that "means" is a bit elusive but one gets used to it after a while.

"Goes to zero" I can handle. "Being zero" makes no sense. And if it's not zero, then there is no infinity because in order to have infinite slices, each slice would need to be zero-wide.

Serendipper wrote: It seems to me that calculus couldn't work if it were based on division by zero, and that if it works, it must not be.

Yes, Berkeley made the same point, calling derivatives the "ghosts of departed quantities." Today we have the theory of limits that makes it all mathematically rigorous; though I agree that it's perhaps not entirely satisfying philosophically.

So how did it work out with Berkeley?

Serendipper wrote:What is it I am failing to see?

The modern theory of limits.

But the limit is just an extrapolation. Instead of saying 1/x =0 where x = infinity, we say 1/x -> 0 where x -> infinity. Limits aren't based on infinity, but purposely avoid infinity by extrapolating towards it. We say "if x gets bigger, then 1/x gets closer to zero".

Serendipper wrote: I mean, it's not an issue of 0/0 making no mathematical sense, it's a philosophical issue of the nonexistence of significance because there is nothing in zero to be significant.

People have been arguing about this since Newton and it's still going on.

I can see why lol

Serendipper wrote:Well, thanks to Hilbert, I've already conceded that the boundless is not the same as the infinite and if it were true that QM required infinity, then no machine nor human mind could model it.

Hilbert space is infinite-dimensional and that's what they use to do QM.

How do they model the infinite part of it?

Serendipper wrote:I applaud your effort, I really do, and I've learned a lot of history because of it

I appreciate your kinds words. it is a FACT that all of our theories of physics require actual infinity. Not unboundedness. Actual infinities. Infinities of points in a continuous space in which to do relativity; infinities of wave functions in an infinite-dimensional Hilbert space to do QM. Nobody has a finitary theory of physics. Nobody.

Well somebody must have a finite theory if computers produce answers.

The electromagnetic force has infinite range, but photons cannot be emitted until their destination is found, which means its range cannot be infinite.

Likewise gravity is an interaction that requires a second something that cannot be infinitely far away.

Serendipper wrote:but I still cannot concede that infinity underpins anything and I'd be lying if I said I could see it.

I wonder if you are still conflating physics (how the world is) with theories of physics (our mathematical models). Could that be the case?

Show me how infinity is used in this y=1/x where x -> infinity. Infinity is never used and the math is not based on infinity. That is the point I'm trying to make and the same point is applicable to QM or anything else that "appears" to be based on infinitary math.

I'd concede that 1+2+3+4+.... = -1/12 is based on infinitary math because of the assumption that 1-1+1-1+1-1+1-1+.... = 1/2. If you reject that is 1/2, then the proof falls apart.

But 1/x where x-> infinity never uses infinity or any strange properties of it.

Serendipper wrote: I'm not being stubborn and feel like I'm walking on eggshells being as amicable and conciliatory as possible in trying not to offend and I'm certainly ready to say "Ooooohhh... I see now", but I just don't see it.

Perfectly ok. I'm easily provoked but I get over it quickly. Your philosophical points are valid but the theories themselves require mathematical infinities. Are you saying you don't believe our theories themselves require mathematical infinity? But they do. I've given as many explanations as I can think of.

I need concrete examples to work with, like 1/x or 1-1+1-1+1...

About the 0/0. Say you're driving in your car and your speedometer says 50mph. That's the derivative of your position function. Conceptually it says that if we divide your distance over time for smaller and smaller intervals, the "instantaneous" change in position is 50. But if we could stop time in that one instant, where is the motion? Back to Zeno again. Ancient mystery. And yet your speedometer has no problem measuring your speed.

If time stopped then there is no motion because motion is relative to time (ie 50 miles per hour). If there is no change in time, then there is no change in distance. The derivative is the slope of the position plotted against time, mx+b, so the constant velocity would be m. No division by zero necessary.

surreptitious75 wrote:When you scold me like that, I feel like I have to walk on eggshells.

I'm an asshole. I apologize. It was completely unwarranted for me to go off on you like that. You were asking good questions about what I wrote. I totally apologize. I am way too thin skinned about imagined slights for my own good.

Also ... since what's an apology without a weasel clause ... I"m still sensitive about all the beatings I took at the hands of Saint. I should really try to let that go!

Serendipper wrote:If reals are not infinite, then primes cannot be infinite, right?

I don't know what that means. The antecedent is false so the implication is valid, but the argument is not sound. It's based on a false premise. Euclid proved that "there is no largest prime" 2000+ years ago and he had no modern math, barely even the concept of number. He didn't prove there are infinitely many of anything. Just that there's no greatest one. Which amounts to the same thing.

Serendipper wrote: So primes are infinite only because reals are infinite and the infinity of primes is just a consequence of the infinity of reals.

Are you familiar with Euclid's beautiful proof?

Serendipper wrote:But what I was saying previously is that we work with the presumed ramifications of infinity, "as if" this or that were infinite, without actually ever using infinity itself. For instance, y = 1/x as x approaches infinity, then y approaches 0, but we don't actually USE infinity in any calculations, but we extrapolate.

Infinity pertains to how we do the math that's used in the calculations of physics. I've agreed to that many times. I make no metaphysical claims about how the world "is" or even if that's a meaningful question. I only say that the math that the physicists use is based on infinity.

Serendipper wrote:Well where is infinity actually used in a calculation?

You will have to ask a physicist. Hilbert space is a function space. Think about a continuous function from the reals to the reals, say. Now think about ALL the continuous functions, all at once. That's like Hilbert space. It's an infinite-dimensional vector space. (Hilbert space has additional restrictions)

Serendipper wrote:I'm going to need some help with this one. If dx = 0, then it contains no information about the change in x, so how can anything result from it? I've always taken dx to mean a differential that is smaller than can be discerned, but still able to convey information.

That's a naive view from the 17th century. Or, it's a perfectly modern view for a physicist or engineer! And it's an ongoing philosophical issue.

However in modern standard math, there are no infinitesimals and what you said is not true. They're not small thingies with discernable qualities.

Serendipper wrote:"Goes to zero" I can handle. "Being zero" makes no sense. And if it's not zero, then there is no infinity because in order to have infinite slices, each slice would need to be zero-wide.

The formal theory of limits resolves this problem.

Serendipper wrote: It seems to me that calculus couldn't work if it were based on division by zero, and that if it works, it must not be.

Did they teach you about limits? If you didn't take calculus or if they taught limits badly (very common) one would not necessarily understand that the modern theory of limits has solved this conceptual problem. It took 200 years.

Serendipper wrote:What is it I am failing to see? I mean, it's not an issue of 0/0 making no mathematical sense, it's a philosophical issue of the nonexistence of significance because there is nothing in zero to be significant.

The modern theory of limits. It solves this ancient problem. It's one of the greatest intellectual achievements of humanity. I wish they explained all this better to students but they don't and that's that. From Newton to Zermelo, the arithmetization of analysis. Putting continuous math completely on the back of logic and the axioms of set theory.

Understand, I'm not saying this is all perfect or that there aren't philosophical and even mathematical objections. I'm only describing what actually happened, and how most people think today. Tomorrow it could all be different. I am not being dogmatic; but I am being accurate in the history and in describing how mainstream mathematicians see calculus today.

This isn't a good venue for this discussion, perhaps a different thread? We can talk about the theory of limits. The point though is that it's not just an extrapolation or gimmick. If that's all it were you'd be right. The big deal with limits is this:

We can start from the rules of predicate logic; assume a handful of set-theoretic axioms; and develop a comprehensive theory of limits that makes calculus and all its generalizations perfectly rigorous. The generalizations like differential geometry and functional analysis are the mathematical heart of modern physics.

Serendipper wrote:Well, thanks to Hilbert, I've already conceded that the boundless is not the same as the infinite and if it were true that QM required infinity, then no machine nor human mind could model it.

You can't argue with the actual historically contingent theory. And like I say, nobody has any idea how to do a finitistic theory that works.

Serendipper wrote:How do they model the infinite part of it?

I think I explained the bit about the space of all continuous functions earlier. That's the model to keep in mind. It's a vector space so it must have a basis. What does a basis look like? That will keep you up nights.

Serendipper wrote:The electromagnetic force has infinite range, but photons cannot be emitted until their destination is found, which means its range cannot be infinite.

Hey you will have to take that up with Maxwell and Einstein and all those big brains. I myself wonder how a photon can travel forever, but I assume the physicists have worked it out.

Serendipper wrote:Likewise gravity is an interaction that requires a second something that cannot be infinitely far away.

Can't disagree. But does it require a second something? Even if there's only one massive body in the universe, its gravity distorts the universe, right?

Serendipper wrote:The other two forces have finite range.

I'm not sure how any of this got directed at me. Am I replying to the wrong post? I am a little confused because you seemed to quote your own post, which I'm replying to. I don't know anything about forces, I'm strictly talking about the historically contingent development of math and physics.

Serendipper wrote:but I still cannot concede that infinity underpins anything and I'd be lying if I said I could see it.

If by anything you mean the physical world, I don't disagree. But if you're talking about the historically contingent theories of physics as they actually developed, you're wrong.

Serendipper wrote:Show me how infinity is used in this y=1/x where x -> infinity. Infinity is never used and the math is not based on infinity. That is the point I'm trying to make and the same point is applicable to QM or anything else that "appears" to be based on infinitary math.

You shouldn't confuse the use of infinity in calculus with the infinite sets used in mathematical physics.

Serendipper wrote:I'd concede that 1+2+3+4+.... = -1/12 is based on infinitary math because of the assumption that 1-1+1-1+1-1+1-1+.... = 1/2. If you reject that is 1/2, then the proof falls apart.

You're looking at some bad Youtube videos. That equation's not true for the usual definition of +. It's about a thing called zeta function regularization. No point going off in that nonproductive direction.

Serendipper wrote:But 1/x where x-> infinity never uses infinity or any strange properties of it.

Everything in science is based on the real numbers and the real numbers are a very strange infinite set.

Serendipper wrote:If time stopped then there is no motion because motion is relative to time (ie 50 miles per hour). If there is no change in time, then there is no change in distance. The derivative is the slope of the position plotted against time, mx+b, so the constant velocity would be m. No division by zero necessary.

If you measured one single instant of time, how would you know that? It's Zeno's arrow. If you stop it in time, it's just sitting there in space. How does it "remember" that it's going forward with a particular velocity?

Well anyway I hope I responded to the right questions. I might have been confused about some of the quoting.

Serendipper wrote:That was an excellent post and qualifies as a treasure to be found on this site

I wish I knew what he is trying to say because all of it is way way above my headI should have studied maths when I was younger but now I am too old and stupidAt least you understand him so he is not completely wasting his time being here

Serendipper wrote:That was an excellent post and qualifies as a treasure to be found on this site

I wish I knew what he is trying to say because all of it is way way above my headI should have studied maths when I was younger but now I am too old and stupidAt least you understand him so he is not completely wasting his time being here

I'll state my thesis as clearly and briefly as possible. No math is required.

* Regarding the world "out there," I take no position and claim no knowledge. There may be a God, there may not be. The world may be discrete or it may be continuous or the question itself may not be meaningful. It may be infinite or finite or the question might not be meaningful. I could be a brain in a vat or a Boltzmann brain: a momentary accidental coherence in a formless and random universe. I take no position, I claim no knowledge, and nothing I say refers to "ultimate reality," assuming again that the phrase is even meaningful at all.

* There are humans, we came out of caves and developed these big brains and our capacity for abstraction. We made up stories about the stars, the seasons, the world. Around the time of the Renaissance our thinking began to become scientific, in the sense that we did repeatable experiments and tried to build mathematical theories about them. This intellectual trend hit its stride with Galileo, and became the dominant way of understanding the world with Newton. It was Galileo who said that the book of nature is written in the language of math. https://www.irishtimes.com/news/science ... -1.3388465

* Since the 17th century, physicists starting with Newton founded their physical theories on infinitary math; that is, math that uses reasoning about infinity. In the next 200 years this infinitary reasoning was placed on a purely logical footing. We can write down some rules of inference, and assume a handful of axioms about sets; and then logically derive all of known mathematics.

* The physicists use this infinitary math to do their most advanced theories. General relativity and quantum physics are based on infinitary math. Our basic idea of time is that of a continuous flow modeled by the mathematical real numbers, which are a highly infinite set (a "larger infinity" than that of the plain old counting numbers).

* Nobody knows if infinitary math is necessarily the foundation of physical science. It's the historically contingent work of flawed people. Tomorrow morning someone could come up with a finitary theory of physics. However nobody has done so yet, and there aren't even any serious efforts to do so. There ARE in fact efforts to base physics on constructive math, which allows infinite objects as long as we can explicitly describe how to construct them. [In standard math there are objects that exist but that we cannot show how to construct].

That's really it. Of the nature of the world, I know nothing. But the history of math and science has led us to the present day, in which we MODEL the world using mathematics that depends on infinite sets. What that means, I don't know. I do personally suspect that just as non-Euclidean geometry started out as a mathematical curiosity yet became the foundation of Einstein's relativity; infinitary mathematics may be telling us something about the world. Something waiting for the next Einstein to discover.

Infinite sets underlie convergence theory, but you can have infinite sets without talking about convergence.

Ecmandu wrote:The odds of you picking a number from infinity, that's in the set, is 100%

Yes.

Ecmandu wrote:The odds of you picking a specific number drawn from the infinite set by someone else, because of convergence, is zero percent.

Not necessarily. It depends on the distribution. For example there is no way to define a probability for picking one set out of the counting numbers 1, 2, 3, .. in such a way that every number has an equal probability. That follows from the axioms for probabilities as set down by Kolmogorov. The requirement of countable additivity implies that no countable set can have a uniform probability measure defined on it.

Ecmandu wrote:So, when you have more than one person, it's impossible for any number to exist!

That doesn't follow from anything you said. I have no idea what you mean.

Ecmandu wrote:This is a fatal flaw in convergence theory.

No it's not. That there is more than one person? The fact that I can't find a parking space because of those pesky other people implies that the mathematical theory of convergence is fatally flawed? No it doesn't.

That's just a matter of definition. The sum of 1/2 + 1/4 + 1/8 + 1/16 + ... IS exactly 1, because that's how the sum of an infinite series is defined. This is taught in the second semester of freshman calculus all over the world.

wtf wrote:Regarding the world out there I take no position and claim no knowledge. There may be a God there may not be. The world may be discrete or it may be continuous or the question itself may not be meaningful. It may be infinite or finite or the question might not be meaningful. I could be a brain in a vat or a Boltzmann brain a momentary accidental coherence in a formless and random universe. I take no position I claim no knowledge and nothing I say refers to ultimate reality assuming again that the phrase is even meaningful at all

I like this because it is entirely non dogmatic about the nature of actual existenceI try to avoid dogmatism in general and especially on something as complex as this

When dealing with infinity, the odds of picking the correct number (that an infinity number generator will pick, or bob next door) isn't 0.0...1. When it converges at infinity, the odds are exactly zero percent.

Convergences work both ways, not only magically adding the infinitesimal, but subtracting it as well.